While the level set method can be used to implicitly represent the location of the interface, to numerically simulate Hele–Shaw flow we are also required to determine the pressure within the viscous fluid, which involves solving a partial differential equation in a complicated domain that changes in time. When applying the boundary integral method for the classic Hele–Shaw problem, the solution to Laplace's equation can be expressed in terms of Green's functions. As such, the problem is reformulated as an integro-differential equation, and nodes need only be placed on the fluid-fluid interface. An alternative choice is to solve for the pressure using the finite difference method, which can be modified to solve problems on complex domains when coupled with level set functions that describe the location of the interface [1]}, [2]}. An advantage of this approach is that the finite difference method can be easily adapted to wide classes of partial differential equations. Further, while the boundary integral method can easily handle non-trivial far-field boundary conditions, their inclusion into the finite difference stencil is not so straightforward. One solution to overcome this is to use a very large computational domain, but this in turn results in significantly longer computational times. Another, more elegant, solution is to use a Dirichlet-to-Neumann map [3]}, which has been shown to accurately capture the far-field boundary condition even when the interface is relatively close to the curve on which the Dirichlet-to-Neumann map is applied [4]}, [5]}, [6]}.
